Subgroups — C₂×C₄×#13133224 #28247496

Order 13568.

3366 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133224	C₂	1	2	yes	maximal
2	C₄×#13133224	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16810579	C₂	1	6	yes	maximal, complemented
4	C₂×C₄×#11719917	C₂	1	3	yes	maximal, complemented
5	C₂²×C₄×#88322	C₂	1	3	yes	maximal, complemented
6	C₂×C₄×#41×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133224	C₄	1	32	yes	direct product factor
8	C₂×#13133224	C₂²	1	16	yes	—
9	C₂²×#11719917	C₂²	1	6	yes	—
10	C₂³×#88322	C₂²	1	6	yes	—
11	#16810579	C₂²	1	48	yes	complemented
12	C₄×#11719917	C₂²	1	24	yes	complemented
13	C₂×#13133128	C₂²	1	6	yes	—
14	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
15	C₂×C₄×#88322	C₂²	1	3	yes	—
16	C₂×C₄×#88322	C₂²	1	6	yes	—
17	C₂×C₄×#88322	C₂²	1	24	yes	complemented
18	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
19	C₄×#41×C₅₃	C₄	1	8	yes	—
20	C₄×#41×C₅₃	C₂²	1	8	yes	—
21	C₂×C₄×Q₈×C₅₃	C₄	1	2	yes	—
22	C₂×C₄×D₄×C₅₃	C₄	1	6	yes	—
23	C₂×#117×C₅₃	C₂²	1	3	yes	—
24	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
25	#13133224	C₂×C₄	1	128	yes	direct product factor
26	C₂×#11719917	C₂×C₄	1	48	yes	complemented
27	C₂×#11719917	C₂³	1	24	yes	—
28	C₂²×#88322	C₂×C₄	1	48	yes	complemented
29	C₂²×#88322	C₂³	1	2	yes	—
30	C₂²×#88322	C₂³	1	6	yes	—
31	C₂²×#88322	C₂³	1	24	yes	—
32	#13133128	C₂³	1	24	yes	—
33	C₄×#88322	C₂³	1	4	yes	—
34	C₄×#88322	C₂³	1	12	yes	—
35	C₄×#88322	C₂³	1	24	yes	—
36	C₂×#41×C₅₃	C₂×C₄	1	16	yes	—
37	C₂×#41×C₅₃	C₂×C₄	1	4	yes	—
38	C₂×#41×C₅₃	C₂³	1	4	yes	—
39	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×D₄×C₅₃	C₂×C₄	1	3	yes	—
41	C₂³×C₄×C₅₃	C₂×C₄	1	6	yes	—
42	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
43	C₄×Q₈×C₅₃	C₂×C₄	1	16	yes	—
44	C₄×D₄×C₅₃	C₂×C₄	1	48	yes	—
45	#117×C₅₃	C₂×C₄	1	12	yes	—
46	#117×C₅₃	C₂³	1	12	yes	—
47	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	6	yes	—
48	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	6	yes	—
49	C₂×C₄²×C₅₃	C₂×C₄	1	6	yes	—
50	C₂×C₄²×C₅₃	C₂×C₄	1	12	yes	—
51	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
52	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
53	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	C₂⁴×C₅₃	C₂²×C₄	1	3	yes	—
55	#41×C₅₃	C₄²	1	32	yes	—
56	#41×C₅₃	C₂²×C₄	1	32	yes	—
57	C₂×Q₈×C₅₃	C₄²	1	8	yes	—
58	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
59	C₂×D₄×C₅₃	C₄²	1	24	yes	—
60	C₂×D₄×C₅₃	C₂²×C₄	1	12	yes	—
61	C₂²×C₄×C₅₃	—	2	24	no	—
62	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	24	yes	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
66	C₂²×C₄×C₅₃	C₂²×C₄	1	24	yes	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
70	C₄⋊C₄×C₅₃	C₂²×C₄	1	24	yes	—
71	C₂²⋊C₄×C₅₃	C₂²×C₄	1	24	yes	—
72	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
73	C₄²×C₅₃	C₂²×C₄	1	6	yes	—
74	C₄²×C₅₃	C₂²×C₄	1	24	yes	—
75	C₄²×C₅₃	C₂⁴	1	2	yes	—
76	C₄²×C₅₃	C₂⁴	1	6	yes	—
77	#11719917	C₂²×C₄	1	96	yes	complemented
78	C₂×#88322	C₂²×C₄	1	8	yes	—
79	C₂×#88322	C₂²×C₄	1	24	yes	—
80	C₂×#88322	C₂²×C₄	1	96	yes	complemented
81	C₂×#88322	C₂⁴	1	4	yes	—
82	C₂×#88322	C₂⁴	1	12	yes	—
83	C₂³×C₅₃	—	2	12	no	—
84	C₂³×C₅₃	C₂×C₄²	1	12	yes	—
85	C₂³×C₅₃	C₂³×C₄	1	6	yes	—
86	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
87	Q₈×C₅₃	C₂×C₄²	1	16	yes	—
88	D₄×C₅₃	C₂×C₄²	1	48	yes	—
89	C₂×C₄×C₅₃	—	2	12	no	—
90	C₂×C₄×C₅₃	—	2	48	no	—
91	C₂×C₄×C₅₃	C₂×C₄²	1	12	yes	—
92	C₂×C₄×C₅₃	C₂×C₄²	1	24	yes	—
93	C₂×C₄×C₅₃	#131	1	8	yes	complemented
94	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
95	C₂×C₄×C₅₃	C₂³×C₄	1	24	yes	—
96	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
97	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
98	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂³×C₄	1	6	yes	—
100	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
101	#88322	C₂³×C₄	1	8	yes	—
102	#88322	C₂³×C₄	1	24	yes	—
103	C₂×C₄×#131	—	53	1	no	maximal, Sylow
104	C₂²×C₅₃	—	2	24	no	—
105	C₂²×C₅₃	—	2	12	no	—
106	C₂²×C₅₃	C₂²×C₄²	1	12	yes	—
107	C₂²×C₅₃	C₂²×C₄²	1	2	yes	—
108	C₂²×C₅₃	C₂×#131	1	4	yes	—
109	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
110	C₄×C₅₃	—	2	12	no	—
111	C₄×C₅₃	C₂²×C₄²	1	2	yes	—
112	C₄×C₅₃	C₂²×C₄²	1	12	yes	—
113	C₄×C₅₃	C₂×#131	1	8	yes	complemented
114	C₄×C₅₃	C₂⁴×C₄	1	2	yes	—
115	C₂²×#131	—	53	2	no	—
116	C₂×C₄×#41	D₅₃	1	1	yes	characteristic
117	C₄×#131	—	53	16	no	—
118	C₂×#405	—	53	6	no	—
119	C₂×C₄×M₁₆	—	53	3	no	—
120	C₂²×C₄×C₈	—	53	3	no	—
121	C₂×C₅₃	—	2	12	no	—
122	C₂×C₅₃	C₄×#131	1	4	yes	complemented
123	C₂×C₅₃	C₂³×C₄²	1	1	yes	characteristic
124	C₂×C₅₃	C₂²×#131	1	2	yes	—
125	C₂²×#41	C₂×D₅₃	1	1	yes	characteristic
126	C₂×#131	—	53	32	no	—
127	C₂×#131	—	53	16	no	—
128	C₂²×M₁₆	—	53	6	no	—
129	C₂³×C₈	—	53	6	no	—
130	C₄×#41	Dic₅₃	1	8	yes	—
131	C₄×#41	C₂×D₅₃	1	8	yes	—
132	C₂×C₄×Q₈	Dic₅₃	1	2	yes	—
133	C₂×C₄×D₄	Dic₅₃	1	6	yes	—
134	C₂×#117	C₂×D₅₃	1	3	yes	—
135	C₂²×C₄²	C₂×D₅₃	1	3	yes	—
136	#405	—	53	48	no	—
137	C₄×M₁₆	—	53	24	no	—
138	C₂×#97	—	53	6	no	—
139	C₂×C₄×C₈	—	53	6	no	—
140	C₂×C₄×C₈	—	53	24	no	—
141	C₂×C₄×C₈	—	53	1	no	—
142	C₂×C₄×C₈	—	53	3	no	—
143	C₅₃	C₂×C₄×#131	1	1	yes	characteristic, complemented, Sylow
144	C₂×#41	C₄×D₅₃	1	16	yes	—
145	C₂×#41	C₂×Dic₅₃	1	4	yes	—
146	C₂×#41	C₂²×D₅₃	1	4	yes	—
147	C₂²×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
148	C₂²×D₄	C₂×Dic₅₃	1	3	yes	—
149	C₂³×C₄	C₂×Dic₅₃	1	6	yes	—
150	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
151	#131	—	53	128	no	—
152	C₂×M₁₆	—	53	48	no	—
153	C₂×M₁₆	—	53	24	no	—
154	C₂²×C₈	—	53	48	no	—
155	C₂²×C₈	—	53	2	no	—
156	C₂²×C₈	—	53	6	no	—
157	C₂²×C₈	—	53	24	no	—
158	C₄×Q₈	C₂×Dic₅₃	1	16	yes	—
159	C₄×D₄	C₂×Dic₅₃	1	48	yes	—
160	#117	C₂×Dic₅₃	1	12	yes	—
161	#117	C₂²×D₅₃	1	12	yes	—
162	C₂×C₄⋊C₄	C₂×Dic₅₃	1	6	yes	—
163	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	6	yes	—
164	C₂×C₄²	C₂×Dic₅₃	1	6	yes	—
165	C₂×C₄²	C₂×Dic₅₃	1	12	yes	—
166	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
167	C₂×C₄²	C₂²×D₅₃	1	12	yes	—
168	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
169	#97	—	53	24	no	—
170	C₄×C₈	—	53	4	no	—
171	C₄×C₈	—	53	12	no	—
172	C₄×C₈	—	53	24	no	—
173	C₂⁴	C₂²×Dic₅₃	1	3	yes	—
174	#41	C₄×Dic₅₃	1	32	yes	—
175	#41	C₂×C₄×D₅₃	1	32	yes	—
176	C₂×Q₈	C₄×Dic₅₃	1	8	yes	—
177	C₂×Q₈	C₂²×Dic₅₃	1	4	yes	—
178	C₂×D₄	C₄×Dic₅₃	1	24	yes	—
179	C₂×D₄	C₂²×Dic₅₃	1	12	yes	—
180	C₂²×C₄	—	2	24	no	—
181	C₂²×C₄	C₂×C₄×D₅₃	1	24	yes	—
182	C₂²×C₄	C₂²×Dic₅₃	1	6	yes	—
183	C₂²×C₄	C₂²×Dic₅₃	1	24	yes	—
184	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
185	C₂²×C₄	C₂²×Dic₅₃	1	6	yes	—
186	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
187	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
188	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
189	M₁₆	—	53	96	no	—
190	C₂×C₈	—	53	8	no	—
191	C₂×C₈	—	53	24	no	—
192	C₂×C₈	—	53	96	no	—
193	C₂×C₈	—	53	4	no	—
194	C₂×C₈	—	53	12	no	—
195	C₄⋊C₄	C₂²×Dic₅₃	1	24	yes	—
196	C₂²⋊C₄	C₂²×Dic₅₃	1	24	yes	—
197	C₄²	C₂²×Dic₅₃	1	2	yes	—
198	C₄²	C₂²×Dic₅₃	1	6	yes	—
199	C₄²	C₂²×Dic₅₃	1	24	yes	—
200	C₄²	C₂³×D₅₃	1	2	yes	—
201	C₄²	C₂³×D₅₃	1	6	yes	—
202	C₂³	—	2	12	no	—
203	C₂³	C₂×C₄×Dic₅₃	1	12	yes	—
204	C₂³	C₂³×Dic₅₃	1	6	yes	—
205	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
206	Q₈	C₂×C₄×Dic₅₃	1	16	yes	—
207	D₄	C₂×C₄×Dic₅₃	1	48	yes	—
208	C₂×C₄	—	2	12	no	—
209	C₂×C₄	—	2	48	no	—
210	C₂×C₄	C₂×C₄×Dic₅₃	1	12	yes	—
211	C₂×C₄	C₂×C₄×Dic₅₃	1	24	yes	—
212	C₂×C₄	#13133224	1	8	yes	direct product factor
213	C₂×C₄	C₂²×C₄×D₅₃	1	4	yes	—
214	C₂×C₄	C₂²×C₄×D₅₃	1	24	yes	—
215	C₂×C₄	C₂³×Dic₅₃	1	12	yes	—
216	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
217	C₂×C₄	C₂³×Dic₅₃	1	6	yes	—
218	C₂×C₄	C₂³×Dic₅₃	1	4	yes	—
219	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
220	C₈	—	53	8	no	—
221	C₈	—	53	24	no	—
222	C₂²	—	2	24	no	—
223	C₂²	—	2	12	no	—
224	C₂²	C₂²×C₄×Dic₅₃	1	12	yes	—
225	C₂²	C₂²×C₄×Dic₅₃	1	2	yes	—
226	C₂²	C₂×#13133224	1	4	yes	—
227	C₂²	C₂⁴×Dic₅₃	1	1	yes	characteristic
228	C₄	—	2	12	no	—
229	C₄	C₂²×C₄×Dic₅₃	1	2	yes	—
230	C₄	C₂²×C₄×Dic₅₃	1	12	yes	—
231	C₄	C₂×#13133224	1	8	yes	direct product factor
232	C₄	C₂³×C₄×D₅₃	1	2	yes	—
233	C₂	—	2	12	no	—
234	C₂	C₄×#13133224	1	4	yes	direct product factor
235	C₂	C₂³×C₄×Dic₅₃	1	1	yes	characteristic
236	C₂	C₂²×#13133224	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₅₃
commutator	C₂×C₅₃	C₂³×C₄²
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂×C₄×#41×C₅₃	C₂
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂²×#41×C₅₃	C₂²
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂×C₄×#41	D₅₃
hypocenter	C₅₃	C₂×C₄×#131
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×#41

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₄×#13133224	C₂	direct (N × H)
3. C₂×#16810579	C₂	semidirect
4. C₂×C₄×#11719917	C₂	semidirect
5. C₂²×C₄×#88322	C₂	semidirect
7. C₂×#13133224	C₄	direct (N × H)
11. #16810579	C₂²	semidirect
12. C₄×#11719917	C₂²	semidirect
17. C₂×C₄×#88322	C₂²	semidirect
25. #13133224	C₂×C₄	direct (N × H)
26. C₂×#11719917	C₂×C₄	semidirect
28. C₂²×#88322	C₂×C₄	semidirect
77. #11719917	C₂²×C₄	semidirect
80. C₂×#88322	C₂²×C₄	semidirect
93. C₂×C₄×C₅₃	#131	semidirect
113. C₄×C₅₃	C₂×#131	semidirect
122. C₂×C₅₃	C₄×#131	semidirect
143. C₅₃	C₂×C₄×#131	semidirect
212. C₂×C₄	#13133224	direct (N × H)
231. C₄	C₂×#13133224	direct (N × H)
234. C₂	C₄×#13133224	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#13133224	2
C₄×#13133224	2
C₂×#16810579	2
C₂×C₄×#11719917	2
C₂²×C₄×#88322	2
C₂×C₄×#41×C₅₃	2
C₂×C₄×#131	53

Subgroups — C2×C4×#13133224 #28247496

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#13133224 #28247496